How can we factor $\displaystyle x^4 + 1$ into the difference of two squares?

[paulmdrdo1]

how to force factor this into the difference of two squares.

$\displaystyle x^4 + 1$

 

[alyafey22]

Re: brushing up on factoring.

\(\displaystyle x^4+1=x^4+2x^2+1-2x^2\)

or

\(\displaystyle x^4+1=x^4-i^2\)

The rest is for you ;)

 

[paulmdrdo1]

Re: brushing up on factoring.

I would get this

$\displaystyle \displaystyle \begin{align*} x^4 + 1 &= x^4 + 2x^2 + 1 - 2x^2 \\& = \left( x^2 + 1 \right) ^2 - \left( \sqrt{2} \, x \right) ^2 \\& = \left( x^2 - \sqrt{2}\, x + 1 \right) \left( x^2 + \sqrt{2}\,x + 1 \right) \end{align*}$

but i want to know what's your reasoning by choosing the term 2x^2?

 

[alyafey22]

Re: brushing up on factoring.

I would get this

$\displaystyle \displaystyle \begin{align*} x^4 + 1 &= x^4 + 2x^2 + 1 - 2x^2 \\& = \left( x^2 + 1 \right) ^2 - \left( \sqrt{2} \, x \right) ^2 \\& = \left( x^2 - \sqrt{2}\, x + 1 \right) \left( x^2 + \sqrt{2}\,x + 1 \right) \end{align*}$

but i want to know what's your reasoning by choosing the term 2x^2?

The idea is complete the square , since if we have for example :

\(\displaystyle x^2+1\) our first glance suggests converting it to $x^2+2x+1$ which is a complete square hence factorizing will be possible .

 

[CellCoree]

Factoring is an important skill in mathematics that allows us to break down a polynomial into smaller, simpler parts. In this case, we are looking at the polynomial $\displaystyle x^4 + 1$. To factor this into the difference of two squares, we need to first identify the perfect square terms within the polynomial. In this case, we have $\displaystyle x^4$, which is a perfect square because it can be written as $\displaystyle (x^2)^2$.

Next, we need to find the square root of this perfect square term, which is $\displaystyle x^2$. Now, we can rewrite our polynomial as $\displaystyle (x^2)^2 + 1$.

To force factor this into the difference of two squares, we can use the formula $\displaystyle a^2 + b^2 = (a + b)(a - b)$. In this case, our $\displaystyle a$ term is $\displaystyle x^2$ and our $\displaystyle b$ term is 1. So, we can rewrite our polynomial as $\displaystyle (x^2)^2 + 1 = (x^2 + 1)(x^2 - 1)$.

Finally, we can simplify this expression further by factoring $\displaystyle x^2 - 1$ into the difference of two squares. This gives us our final factored form:

$\displaystyle x^4 + 1 = (x^2 + 1)(x + 1)(x - 1)$.

By using the difference of two squares formula, we were able to break down the polynomial into smaller, simpler parts that are easier to work with. This skill is useful in solving equations and understanding the behavior of polynomials.